Here's an experiment you can all do, and all you need is a piano. Play C6 and F#6 together, as loudly as you can. If the theory is correct, the resultant between F#6 (1484 Hz) and C6 (1050.5 Hz) will be 433.5 Hz, which is just 25 cents below A4. I tried this, not expecting to hear any resultant. I was quite surprised to hear a faint A4, or something like it. But as I listened, I noticed that the A4 resultant behaved strangely. As soon as the C6 and F#6 began to decay, the A4 resultant suddenly dropped out. It did not gradually decay. And if I play C6 and F#6 together softly, I hear no A4 resultant at all. This leads me to believe that the A4 resultant is not being caused by C6 and F#6 beating together in the usual fashion. If that were the case, the resultant would decay exactly the same as the two primary notes. Instead, I think this is a case of non-linear mixing (as alluded to by Sarah Fox several days ago). When the volume level falls below the threshold for making the distortion or buzzing, then the resultant tone suddenly stops. There are lots of opportunities for non-linear mixing when playing two notes on a piano. No piano is perfectly solid. If there is a loose screw somewhere, or if two parts of the piano are just barely touching, then you will have non-linear mixing. (In the extreme case, these conditions cause noticeable buzzing.) At high volume levels, there can even be non-linear mixing in our ears, when the bones that carry the sound from the eardrum reach the limit of their travel. In fact it is difficult to guarantee a purely linear addition of two tones. Any distortion of the sound causes the component sounds to interact non-linearly, and thus produce real acoustic energy at the resultant frequency. For example, Don Mannino's suggestion: >....Back to your original question, Ric. I made a recording for you, and >you can hear the resultant tone. I made a wav file of 20K and 20.5K >tones, one in each ear. Then I combined them into a single mono file. If >you play it in good headphones and turn up the volume loud, you can just >hear the 500hz tone as a pitch. The process of making wav files includes time-sampling and quantization, both of which are somewhat non-linear. If there is a 500 Hz resultant in Don's wav file, it is probably an artifact of the imperfections of the recording process. If it were possible to record 20 kHz and 20.5 kHz tones and mix them with no distortion at all, then there would be no 500 Hz resultant. Since it is hard to find such a perfect recording system, then the theory can perhaps be tested by finding a poorer recording system - one with more distortion. If my theory is correct, the resultant tone should be even stronger when played on such a system with more distortion. -Bob Scott Ypsilanti, MI
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